Nope. It's like in French where our teacher used to remind us that you go 'to' the butcher shop, but 'at' the brothel (and accessorily 'at' school).
Like 'dul ar dinnéar', its seems connected to occasions where you assume your position (in the class, at the table) in a place where you have your place...
Dul ar an bhfara: to roost.
Dul ar garda: to mount guard.
Dul ar pósadh: to go to a wedding.
Mind you, you can opt for 'dul chun na scoile' if you wanted to...

Interesting! In the same series I read "go dti an bhialann". Does your logic apply here to: would you say "ar bialann" when you had reserved a table? Or is the rule you refer to here to places where you usually have a place?

Yes, I'd like to know the same - I was translating into Irish and used seo, as I thought without it, the sentence could have meant far in the past - like something you'd ask your parents or grandparents.

If you want to be specific about referring to "this morning", the Irish is ar maidin inniu - ar maidin seo is considered béarlachas.

But unless context or tense implies otherwise, ar maidin is read as "this morning". If you are clearly talking about a day in the past, or using the habitual present or past tenses, "in the morning(s)" makes more sense.